Hardness of approximation, aka inapproximability.

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Petrank's proof for the APX-hardness of MAX k-VERTEX COVER on subcubic graphs

My question is about the following maximization problem, which is the "fixed cardinality" version of MIN VERTEX COVER. I am interested in the restriction to subcubic graphs (i.e. of maximum degree 3). ...
9
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93 views

NP-hardness of approximation for unconstrained submodular maximization

The problem of unconstrained submodular maximization can be phrased as follows: Given a non-negative submodular function $f$ on a domain $D$ find a set $S \subseteq D$ maximizing $f(S)$. Here a ...
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70 views

About representability of optimization versions of NP-complete questions as polynomial optimization over the hypercube

Naively, in my very limited awareness, it feels that the Max-CUT is a very "special" NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a ...
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50 views

About complexity of recovering or learning Bayesian networks

Are there complexity theoretic results about recoverability or learnability of the marginals (of the source vertices) and the conditionals (along each of the edges) of a Bayesian network from having ...
4
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2answers
109 views

Finding a minimal context free grammar that recognizes a finite set of strings of bounded length

Problem: Given a finite set of strings $\{x_1, x_2, ..., x_n\}$ of length $\ell$ or less from some finite alphabet $\Sigma=\{a_1, a_2, ..., a_k\}$, find the minimal context free grammar that ...
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40 views

Thresholding technique for Bottleneck Optimization Problems like k-centre, k-suppliers etc

There is a well know thresholding techniques (introduced by Hochbaum and Shmoys in [1]) for bottleneck optimization problems which preprocesses the instance which is a graph whose edges satisfy the ...
5
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1answer
126 views

Maximizing a monotone supermodular function s.t. cardinality

I've tried to comb the literature and seen a lot of references to results that almost but don't quite seem to address this. Question: Is it known to be true or is there a hardness result ...
5
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230 views

Maximizing the number of selected edges with opposing requirements

Consider the following problem: Input: a complete bipartite graph $G$ with its edges colored either white or black, a number $k$. Output: a subset of vertices $W$ of size $k$ which maximizes the ...
3
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1answer
194 views

Approximation algorithms for the maximum $2$-independence set problem

I am interested in approximating the maximum $2$-independent set problem in arbitrary graphs. In a graph $G$ a set $I$ of vertices is called $2$-independent if the distance between any two distinct ...
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31 views

On approx-preserving P- and A-reducibilities

Let $X$ and $Y$ be two NPO problems. Let $(f,g)$ be a reduction between $X$ and $Y$, in particular, assume that $(f,g)$ is both P-reduction and A-reduction, i.e., there exist two poly-time ...
3
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2answers
281 views

Graph coloring/partitioning problem

I'm interested in the complexity of the following problem: Problem $P$: Given an undirected planar graph $G=(V,E)$ and a weight function $w: E \rightarrow \mathbb{Z}$ (so weights can be negative, ...
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267 views

smallest circuit size using XOR gates

Suppose we are given a set of n boolean variables x_1,...,x_n and a set of m functions y_1...y_m where each y_i is the XOR of a (given) subset of these variables. The goal is to compute the minimum ...
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43 views

hardness of approximating clique: how using FGLSS reduction with PCP verifier of hastad

I try to understand the $n^{1-\epsilon}$ hardness of approximating clique for any $\epsilon$ provided in [1]: www.nada.kth.se/~johanh/cliqueinap.ps In fact, I only want to understand the proof of ...
3
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0answers
58 views

Set cover approximation ratio as a function of m (number of sets)

Feige's well known result (and more recent results) show that set cover cannot be approximated within a factor of $(1 - o(1)) \ln n$, where $n$ is the number of variables. What if we want an ...
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80 views

Is there any research on approximation of reals with computable numbers

I was wondering if there is any research in the field of Diophantine Approximation using the computable numbers. It seems to be a good fit, a dense countable set with a variety of different potential ...
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65 views

Hardness of approximately counting independent sets with a PRAS, rather than FPRAS

It is known that approximately counting the independent sets of a graph is hard, even if randomness can be used, and even if we restrict ourselves to bounded degree graphs with degree bound at least ...
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1answer
280 views

What are multiple rounds of SOS/Lasserre hierarchy?

Is that the same as saying the one will try to generate a higher-degree "pseudo expectation functional" by solving a SOS-program ? Or is there a difference between the two things? Or to take a ...
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1answer
264 views

The Goemans-Williamson algorithm in the $SOS$ framework

If there is a variable $x_i$ for every vertex $i$ of a $d$-regular graph $G$ then assigning $x_i = \pm 1$ gives a cut, say $(S,\bar{S})$, of the graph. We can then see that, $\langle x,L x\rangle$, ...
3
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1answer
201 views

What is a “level-r pseudo expectation functional”?

In the context of the SOS hierarchy papers, it seems that a "level-r psuedo expectation functional" is the same as an operator taking expectations of functions just that this one has the restriction ...
5
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160 views

Is MAX-SAT SETH (like) hard?

If we make a random assignment to the variables in $k$-sat ($m$ clauses), we are going to satisfy $(1-2^{-k})m$ clauses in expectation. In general satisfying fewer clauses is considered easy. There ...
3
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1answer
122 views

Set cover in which some pairs of sets are forbidden

I'm trying to find an approximation algorithm for a variant of the weighted set cover problem. However, this variation doesn't seem to let me apply the traditional set cover arguments for an ...
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76 views

$\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ problems

All the problems which are either $\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ cannot admit a $\mathsf{PTAS}$, unless $P=NP$. I would like to know whether $\mathsf{MaxSNP}$ ...
3
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2answers
203 views

What is the relationship between $\mathsf{APX}$ and $\mathsf{MaxSNP}$ classes?

My understanding of these classes is a really fuzzy. The more I am trying to read the more I am getting confused. Can anyone help me understand the relationship between these classes. More precisely, ...
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1answer
378 views

Does $NP$-hardness of $c$-approximation (for some $c>1$) imply $APX$-hardness?

Assume that for a given minimization problem with only integer solutions, it is $NP$-hard to decide if the optimal solution is 5 or 6. I.e., a polynomial-time algorithm with an approximation ratio ...
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40 views

APx hardness of Multiterminal Cut Problem

In a Multiterminal Cut problem input is a graph G=(V,E) and a set of k terminals T which is a subset of vertex set V. There is a weight w(e) associated with each edge in the graph. The question is to ...
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97 views

The distribution on the solution space induced by randomized rounding

Consider the Goemans-Williamson algorithm for the MAX-CUT problem. It is known, that if $maxcut(G) \geq 1-\epsilon$, then the algorithm returns a cut $S$ of fractional size at least ...
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1answer
126 views

Confusion in 2012 paper by Austrin and Håstad regarding hardness of approximating GLST

The paper in question is "On the Usefulness of Predicates", Per Austrin, Johan Håstad (arXiv:1204.5662 [cs.CC]). On page 13, Example 8.2 they define a predicate $P$ which is $GLST$ with an ...
4
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1answer
60 views

Definition of Projection Measure in the characterization of strong approximation Resistance in a paper by Khot et al

I'm reading a paper about Constraint Satisfaction Problems, specifically "A Characterization of Strong Approximation Resistance", Subhash Khot, Madhur Tulsiani, Pratik Worah (ECCC TR13-075). The ...
9
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1answer
238 views

A purely graph-theoretic explanation of the reduction from Unique Label Cover to Max-Cut

I am studying the Unique Games Conjecture and the famous reduction to Max-Cut of Khot et al. From their paper and elsewhere on the internet, most authors use (what to me is) an implicit equivalence ...
4
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1answer
136 views

Inapproximability of $(\alpha, \beta)$ bi-criteria approximation

An $(\alpha, \beta)$ bi-criteria approximation algorithm for $k$-center is defined as an algorithm that returns a solution whose value is $\beta \cdot OPT$ ($OPT$ being the optimal solution for the ...
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45 views

inapproximability of logarithic factor of indepence set [closed]

The hardness result derived using PCP theorem for Independent set suggests that there exists some absolute constant $\epsilon_0$ such that for $0< \epsilon < \epsilon_0$, it is hard to ...
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328 views

Is graph coloring complete for poly-APX?

Is the graph coloring problem complete for poly-APX under C-reductions (alternatively, under AP-reductions)? For the graph coloring problem, speaking of a feasible solution means a proper coloring for ...
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52 views

How can I show that zero-one programming is not in APX?

How can I show that zero-one programming is not in APX? Vertex Cover Problem is in the APX class. So can I try a PTAS-reduction from the zero-one programming problem to Vertex Cover and show that ...
9
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2answers
217 views

Approximating #P-hard problems

Consider the classical #P-complete problem #3SAT, i.e., to count the number of valuations to make a 3CNF with $n$ variables satisfiable. I am interested in the additive approximability. Clearly, there ...
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1answer
183 views

Complexity of linear programming

It is known that Linear Programming (LP) is P-complete. I am interested in approximation algorithms for LP. There are numerous inapproximability results for NP optimization problems, e.g. it is ...
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140 views

Calculating exact/approximate solution to a formula

Suppose we have a set of variable $\mathbf{y} = \left(y_1, ..., y_n \right)$. Also consider the set of functions $g_i(y_i), 1 \leq i \leq n$. Note that $g_i()$ is dependent only on $y_i$. Consider ...
4
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1answer
135 views

Hardness of approximating chromatic number of triangle-free graphs

The chromatic number of graph, $\chi( G)$ is hard to approximate for general graphs. Are there results of hardness of approximating $\chi(G)$ for triangle-free graphs?
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147 views

Approximate c-chromatic number, each color class is P4-free (cograph)

The classic chromatic number of graph, $\chi(G)$, describes the minimum number of colors needed so that each color class is an independent set. There are many other graph coloring definitions. One of ...
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36 views

Approximation algorithm to a minimization problem with optimization function going to zero

I have a simple function to minimize, but it is not discrete: $f(x) = 2x - \alpha x - c(x)$, where $x$ is a natural number, $\alpha$ is a rational number, $0 \leq \alpha \leq 1$, and $0 \leq c(x) \leq ...
14
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1answer
390 views

Does PSPACE-completeness imply approximation hardness?

It is mentioned in a comment in another cstheorySE post that PSPACE-completeness imply APX-hardness. Can anyone please explain/share a reference for it? Is this "tight"? (i.e., are there ...
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3answers
607 views

Planted Clique in G(n,p), varying p

In the planted clique problem, one must recover a $k$-clique planted in an Erdos-Renyi random graph $G(n,p)$. This has mostly been looked at for $p=\frac{1}{2}$, in which case it is known to be ...
3
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2answers
185 views

Set packing with maximum coverage objective

We are given a universe $\mathcal{U}=\{e_1,..,e_n\}$ and a set of subsets $\mathcal{S}=\{s_1,s_2,...,s_m\}\subseteq 2^\mathcal{U}$. Set-Packing asks how many disjoint sets we can pack, and is defined ...
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1answer
71 views

Bellman principle and approximability

Does anybody know if a combinatorial optimzation problem that enjoys the Bellman's optimality principle can in automatic way be approximated?
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231 views

Comparing graphs

I am looking for a kick in the right direction. What i am trying to do is to compute "similarity" between two graphs, where I define "similarity" as the number of shared paths. example: ...
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1answer
215 views

Almost always almost right

I am looking for a complexity class that relates to APX as BPP relates to P. I have already asked the same question here, but perhaps TCS would be a more fruitful location for answers. The reason for ...
4
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2answers
368 views

Variants of Densest Subgraph Problems

Given a graph $G$, and a vertex-induced subgraph $H$ of $G$, there are two superficially-similar definitions ways to define the density of $H$: (1) Average degree of vertices in $H$ (2) What I will ...
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2answers
277 views

Approximation in subexponental time

There are studies about approximation algorithms for NP complete problems in Polynomial time and exact algorithms in exponential time. Are there studies about approximation algorithms for NP complete ...
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0answers
43 views

Approximating BLEDP on restricted graph classes

In the edge-disjoint paths (EDP) problem, we are given an undirected graph $G$, and a set $\{ (s_i,t_i) \mid 1 \leq i \leq k \}$ of $k$ source-sink pairs. The objective is to maximize the number of ...
3
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1answer
103 views

Approximation algorithms for multicut for special classes of graphs

The multicut problem is the following. Given a graph $G=(V,E)$ with edge costs $c_e$ for edge $e$ and a set of $k$ terminal pairs $\{(s_1,t_1),\ldots,(s_k,t_k)\}$, the objective is to find a set of ...
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175 views

The non-metric k-median problem

It is well-known that the non-metric $k$-median problem cannot be approximated better than $O(\log(n))$ (by a gap preserving reduction from the set cover problem). Is there any logarithmic ...